Question: find all singular points of function $f(z)=|z|^2+\frac{1}{z}$
My attempt:
Since we can be written as, $f(z)=z\bar{z}+\frac{1}{z}$
$$→\frac{∂f}{∂\bar{z}}=z$$
$$=0 \text{ if and only if }z=0$$
So it follows that, $f$ is not differentiable when $z≠0$(is am I correct?)
What about $z=0$ ? Is it singular point ? Further, what types of singularities $f$ have?
Please help me.... stuck on it :-(
If a function is holomorphic, then $\frac{∂f}{∂\overline{z}}=0$ so taking the contrapositive gives you that f is not differentiable (thus not analytic) when $z\neq 0$. If you look closely at your function, $1/z$ blows up when $z=0$, hence $z=0$ is also a singularity and your function has singularites $\forall z\in \mathbb{C}$.